One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

摘要

We characterize one-sided weighted Sobolev spaces W-1,W-P(R, omega), where w is a one-sided Sawyer weight, in terms of a.e. and weighted L-P limits as alpha -> 1(-) of Marchand fractional derivatives of order alpha. Similar results for weighted Sobolev spaces W-2,W-p (R-n, nu), where v is an A(p)-Muckenhoupt weight, are proved in terms of limits as s -> 1(-) of fractional Laplacians (-Delta)(s). These are Bourgain-Brezis- Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e. and weighted L-P limits as alpha, s -> 0(+). (C) 2019 Elsevier Ltd. All rights reserved.